EXPONENTS, ATTRACTORS AND HOPF DECOMPOSITIONS FOR INTERVAL MAPS

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following: (1) There is a set C(T) such that the omega-limit of Lebesbue-a.e. point equals C(T). C(T) is a finite union of closed intervals or it coincides with the closure of the critical orbit. (2) There is a constant-lambda-T such that lambda-T = limBAR(n --> infinity) 1/n log \(T(n))'(x)\ for Lebesgue-a.e. x. (3) lambda-T > 0 if and only if T has an absolutely continuous invariant measure of positive entropy. (4) lambda-T greater-than-or-equal-to inf {p-1 log \(T(p))'(z)\: T(p)Z = Z}, i.e. uniform hyperbolicity on periodic points implies-lambda-T > 0, and-lambda-T < 0 implies the existence of a stable periodic orbit.